Solving $5 \frac{2}{3} + \frac{29}{69} + 6 \frac{21}{23}$ A Step-by-Step Guide

In the realm of mathematics, mastering arithmetic operations with mixed numbers and fractions is a foundational skill. This article delves into a comprehensive, step-by-step approach to solving the problem $5 \frac{2}{3} + \frac{29}{69} + 6 \frac{21}{23}$, ensuring that the final answer is reduced to its simplest form. We will explore the essential techniques required to tackle such problems effectively. This includes converting mixed numbers to improper fractions, finding a common denominator, performing the addition, and simplifying the result. By understanding these concepts, you will be well-equipped to confidently handle a variety of similar mathematical challenges. The ability to manipulate fractions and mixed numbers is crucial not only in academic settings but also in practical, everyday situations, such as cooking, measuring, and financial calculations. Let's embark on this mathematical journey to enhance your skills and build a solid foundation in arithmetic.

Understanding Mixed Numbers and Improper Fractions

To effectively solve the problem, it's crucial to understand the nature of mixed numbers and improper fractions. A mixed number is a combination of a whole number and a proper fraction (where the numerator is less than the denominator), such as $5 \frac{2}{3}$ and $6 \frac{21}{23}$. On the other hand, an improper fraction is a fraction where the numerator is greater than or equal to the denominator, such as $\frac{5}{3}$. The first step in adding mixed numbers is often to convert them into improper fractions. This conversion makes it easier to find a common denominator and perform the addition. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator of the fraction and then add the numerator. The result becomes the new numerator, and the denominator stays the same. For instance, to convert $5 \frac{2}{3}$ to an improper fraction, we calculate $(5 imes 3) + 2 = 17$, so the improper fraction is $\frac{17}{3}$. Similarly, $6 \frac{21}{23}$ becomes $\frac{(6 imes 23) + 21}{23} = \frac{159}{23}$. Once we have converted all mixed numbers to improper fractions, we can proceed with the addition process. This conversion is a fundamental step in simplifying complex arithmetic problems involving mixed numbers and fractions.

Step 1 Converting Mixed Numbers to Improper Fractions

The initial crucial step in solving the problem $5 \frac{2}{3} + \frac{29}{69} + 6 \frac{21}{23}$ is to convert the mixed numbers into improper fractions. This conversion simplifies the addition process and allows us to work with a common fractional form. Let's begin by converting $5 \frac{2}{3}$ into an improper fraction. To do this, we multiply the whole number (5) by the denominator (3) and add the numerator (2). This gives us $(5 imes 3) + 2 = 15 + 2 = 17$. So, the improper fraction equivalent of $5 \frac{2}{3}$ is $\frac{17}{3}$. Next, we convert $6 \frac{21}{23}$ into an improper fraction. We multiply the whole number (6) by the denominator (23) and add the numerator (21). This gives us $(6 imes 23) + 21 = 138 + 21 = 159$. Thus, the improper fraction equivalent of $6 \frac{21}{23}$ is $\frac{159}{23}$. Now that we have converted both mixed numbers into improper fractions, the original problem can be rewritten as $\frac{17}{3} + \frac{29}{69} + \frac{159}{23}$. This transformation sets the stage for the next step: finding a common denominator to add these fractions effectively. Converting mixed numbers to improper fractions is a foundational technique that simplifies the addition and subtraction of fractions.

Step 2 Finding the Least Common Denominator (LCD)

After converting the mixed numbers to improper fractions, the next critical step is to find the Least Common Denominator (LCD). The LCD is the smallest common multiple of the denominators of the fractions we are adding. In our problem, we have the fractions $\frac{17}{3}$, $\frac{29}{69}$, and $\frac{159}{23}$. The denominators are 3, 69, and 23. To find the LCD, we need to determine the smallest number that is divisible by all three denominators. First, let's list the prime factors of each denominator: 3 is a prime number, so its prime factor is simply 3. For 69, the prime factors are $3 imes 23$. For 23, which is also a prime number, the prime factor is 23. Now, we identify the highest power of each prime factor that appears in any of the denominators. We have 3 appearing once (in 3 and 69) and 23 appearing once (in 69 and 23). Therefore, the LCD is the product of these prime factors: $3 imes 23 = 69$. So, the LCD for the fractions $\frac{17}{3}$, $\frac{29}{69}$, and $\frac{159}{23}$ is 69. Finding the LCD is crucial because it allows us to rewrite the fractions with a common denominator, making addition straightforward. This step ensures that we are adding fractions with comparable parts, which is essential for obtaining the correct sum.

Step 3 Rewriting Fractions with the LCD

Now that we have determined the Least Common Denominator (LCD) to be 69, the next step is to rewrite each fraction with this common denominator. This involves adjusting the numerators of the fractions while keeping the values equivalent. For the first fraction, $\frac{17}{3}$, we need to find what number we multiply 3 by to get 69. Since $69 ext{ ÷ } 3 = 23$, we multiply both the numerator and the denominator of $\frac{17}{3}$ by 23. This gives us $\frac{17 imes 23}{3 imes 23} = \frac{391}{69}$. The second fraction, $\frac{29}{69}$, already has the denominator 69, so we don't need to change it. It remains as $\frac{29}{69}$. For the third fraction, $\frac{159}{23}$, we need to find what number we multiply 23 by to get 69. Since $69 ext{ ÷ } 23 = 3$, we multiply both the numerator and the denominator of $\frac{159}{23}$ by 3. This gives us $\frac{159 imes 3}{23 imes 3} = \frac{477}{69}$. Now, our original problem is rewritten with all fractions having the common denominator of 69: $\frac{391}{69} + \frac{29}{69} + \frac{477}{69}$. Rewriting fractions with a common denominator is a key step in adding or subtracting fractions, as it ensures that we are combining like parts. This step sets the stage for the actual addition of the fractions, which will be our next focus.

Step 4 Adding the Fractions

With all the fractions now having a common denominator of 69, we can proceed with the addition. Our problem is now expressed as $\frac{391}{69} + \frac{29}{69} + \frac{477}{69}$. To add fractions with a common denominator, we simply add the numerators and keep the denominator the same. So, we add the numerators: $391 + 29 + 477$. Adding these numbers together, we get $391 + 29 = 420$, and then $420 + 477 = 897$. Therefore, the sum of the numerators is 897. The denominator remains 69, so the sum of the fractions is $\frac{897}{69}$. This fraction represents the total of the three fractions we started with. However, it is not yet in its simplest form. The next step is to simplify the fraction, if possible, by reducing it to its lowest terms. This involves finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by it. Simplifying the fraction is crucial to expressing the answer in its most concise form. In the next step, we will address the process of simplifying the fraction $\frac{897}{69}$.

Step 5 Simplifying the Improper Fraction

After adding the fractions, we obtained the improper fraction $\frac{897}{69}$. Now, it's essential to simplify this fraction to its lowest terms. Simplifying a fraction involves dividing both the numerator and the denominator by their greatest common divisor (GCD). First, we need to find the GCD of 897 and 69. The prime factorization of 69 is $3 imes 23$. Now, let's check if 897 is divisible by either 3 or 23. To check divisibility by 3, we add the digits of 897: $8 + 9 + 7 = 24$. Since 24 is divisible by 3, 897 is also divisible by 3. Dividing 897 by 3, we get 299. Now we have $\frac{897}{69} = \frac{3 imes 299}{3 imes 23}$. We can cancel out the common factor of 3, which gives us $\frac{299}{23}$. Next, we check if 299 is divisible by 23. Dividing 299 by 23, we get 13. So, $299 = 23 imes 13$. Thus, the fraction becomes $\frac{23 imes 13}{23}$. We can cancel out the common factor of 23, leaving us with $\frac{13}{1}$, which is simply 13. Therefore, the simplified form of the improper fraction $\frac{897}{69}$ is 13. This simplification process is crucial to expressing the final answer in its most concise and understandable form. In the next step, we will present the final answer.

Final Answer

After meticulously performing each step, we have arrived at the final answer. We started with the problem $5 \frac{2}{3} + \frac{29}{69} + 6 \frac{21}{23}$. We converted the mixed numbers to improper fractions, found the least common denominator, rewrote the fractions with the LCD, added the fractions, and then simplified the resulting improper fraction. The final simplified answer is 13. This process demonstrates the importance of each step in solving complex fraction problems. By converting mixed numbers to improper fractions, finding a common denominator, and simplifying the result, we ensure accuracy and clarity in our solution. This methodical approach is applicable to a wide range of fraction and mixed number problems, making it a valuable skill in mathematics. Understanding these steps not only helps in solving mathematical problems but also enhances problem-solving skills in various real-life scenarios. Therefore, mastering these techniques is essential for building a strong foundation in mathematics.

This comprehensive guide has walked you through the process of adding mixed numbers and fractions, ensuring the final answer is in its simplest form. Remember to practice these steps to build confidence and proficiency in solving similar problems. This methodical approach will serve as a valuable tool in your mathematical journey.